STEP-BY-STEP LESSON · §9.1
Introduction to Probability
Distinguish outcomes, events, and probabilities.
Before you begin
Sets and membership
A set records which objects belong to it. Repetition and listing order do not change a set. x∈A says that x is an element; A⊆B says every element of A is also in B.
Fractions
a/b is a quotient with b≠0. For integer a,b it represents a rational number. When adding fractions, put them over a common nonzero denominator; keep exact values until the final result.
Symbols
- S
- sample space
- E⊆S
- event: a set of outcomes
- P(E)
- probability of E
- |S|
- Number of elements in finite set S
- S\E
- Elements of S outside E
Definitions and notation for this topic
Sample space
S; Ω
The set of possible outcomes of a random experiment.
This is a tutor example.
The current probability lessons and Epp use S; Ω is another common name. In Ω(g), the same Greek letter instead belongs to asymptotic growth notation.
Separate glossary example
Rolling a six-sided die: Ω={1,2,3,4,5,6}.
Event
A⊆S
A set of outcomes for which the event occurs.
An event may contain several outcomes.
Separate glossary example
An even die result: A={2,4,6}.
Probability
P(A)
A numerical measure of how likely an event is, between 0 and 1, assigned by the probability model.
Counting favorable outcomes and dividing by the total requires finitely many equally likely outcomes.
The formula P(A)=|A|/|S| requires a finite sample space with equally likely outcomes; it is not a general definition for every model.
Separate glossary example
For a fair die, P({2,4,6})=3/6.
A finite probability model
P(A)=∑_{ω∈A}p_ω; ∑_{ω∈S}p_ω=1On a finite nonempty sample space S, assign each outcome ω a weight p_ω≥0 with total 1. The probability of event A⊆S is the sum of its outcome weights.
This gives P(∅)=0, P(S)=1 and additivity for disjoint events. The counting shortcut |A|/|S| is valid only when every outcome has weight 1/|S|.
Separate glossary example
If S={a,b,c} has weights 1/2,1/3,1/6, then P({a,b})=5/6, not 2/3.
Step by step
Step 1 / 6
Name an outcome
An outcome is one fully specified result of an experiment. For one die, it is a number from 1 to 6.
A clear experiment is needed before counting.
Worked example
For a fair die, find the probability of a result greater than 4.
- The die has six faces with equal probability 1/6 each.
- S={1,2,3,4,5,6}; E={5,6}.
- There are 6 outcomes and 2 favorable ones; P(E)=2/6=1/3.
Optional self-check
For two independent fair dice, are sums 2 and 7 equally likely? Explain by counting ordered pairs.
Show answer
No. Sum 2 has only (1,1), while sum 7 has six pairs (1,6) through (6,1). Their probabilities are 1/36 and 6/36.